France World Cup winner Thauvin sent a woeful spot-kick wide in the first half and Swedish winger Nicolas Eliasson then converted twice from close range after the break to give the away side an unassailable lead behind closed doors at the Velodrome.
Argentine forward Dario Benedetto pulled one back late on for Andre Villas-Boas's side but this result left them with just one win in six Ligue 1 games.
Also beaten by Paris Saint-Germain in the Champions Trophy in midweek, Marseille are sixth in the league, now eight points behind leaders Lyon, albeit still with a game in hand.
Nimes move off the foot of the table after their first win in nine, but they remain in the bottom three.
Lyon are at home to Metz on Sunday, but Paris Saint-Germain -- a point off the top in second -- can move provisionally top by winning at Angers later on Saturday.
PSG will be without their new coach Mauricio Pochettino, who is isolating after it was revealed on Friday that he had tested positive for coronavirus.
Meanwhile relegation-threatened Lorient's home game against fellow strugglers Dijon has been postponed after a Covid-19 outbreak in the Brittany side.